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The inequality given is:

\[
\frac{(x - 7)(x + 5)}{x} \geq 0
\]

Let's solve this algebraically by following these steps:

### Step 1: Identify critical points
The expression involves factors \((x - 7)\), \((x + 5)\), and \(x\). The critical points are the values of \(x\) that make any factor equal to zero:

1. \(x = 0\) (from the denominator).
2. \(x = 7\) (from \((x - 7) = 0\)).
3. \(x = -5\) (from \((x + 5) = 0\)).

So, the critical points are \(x = -5\), \(x = 0\), and \(x = 7\).

### Step 2: Determine the sign in each interval
We now test the sign of the expression in the intervals determined by the critical points: \((- \infty, -5)\), \((-5, 0)\), \((0, 7)\), and \((7, \infty)\).

- For \(x \in (-\infty, -5)\): Choose \(x = -6\):
  \[
  \frac{(-6 - 7)(-6 + 5)}{-6} = \frac{(-13)(-1)}{-6} = \frac{13}{6}, \quad \text{positive}.
  \]

- For \(x \in (-5, 0)\): Choose \(x = -1\):
  \[
  \frac{(-1 - 7)(-1 + 5)}{-1} = \frac{(-8)(4)}{-1} = 32, \quad \text{positive}.
  \]

- For \(x \in (0, 7)\): Choose \(x = 1\):
  \[
  \frac{(1 - 7)(1 + 5)}{1} = \frac{(-6)(6)}{1} = -36, \quad \text{negative}.
  \]

- For \(x \in (7, \infty)\): Choose \(x = 8\):
  \[
  \frac{(8 - 7)(8 + 5)}{8} = \frac{(1)(13)}{8} = \frac{13}{8}, \quad \text{positive}.
  \]

### Step 3: Analyze the critical points
- At \(x = -5\), the numerator becomes zero, so \(\frac{(x - 7)(x + 5)}{x} = 0\). This satisfies the inequality.
- At \(x = 0\), the expression is undefined (division by zero).
- At \(x = 7\), the numerator becomes zero, so the expression equals zero, satisfying the inequality.

### Step 4: Conclusion
The solution to the inequality is where the expression is either positive or zero. Based on the sign analysis and critical points, the solution in interval notation is:

\[
(-\infty, -5] \cup (0, 7]
\]

Would you like a more detailed explanation on any step or have further questions?

Here are 5 related questions for deeper understanding:
1. How would the solution change if the inequality were strict (i.e., \( > 0 \) instead of \( \geq 0 \))?
2. What happens if one of the critical points were a repeated root?
3. How can we visualize this inequality on a number line?
4. Why is \(x = 0\) excluded from the solution?
5. How do rational inequalities differ from polynomial inequalities?

**Tip:** Always check for undefined points in rational inequalities to avoid errors in your solution!

Solve the inequality algebraically: (x-7)(x+5)/x ≥ 0.

Learn how to solve the rational inequality (x-7)(x+5)/x ≥ 0 step-by-step. This guide covers key algebraic concepts such as critical points, sign analysis, and rational inequalities. Suitable for high school students, Grades 10-12.

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